Previously optical wave analysis methods such as the Finite Element Method (FEM) or the ‘Beam Propagation Method’ (BPM) have been available for calculating ray propagation in dielectric channel waveguides, especially optical waveguides. However these methods can only be used efficiently if only one mode or a few modes are to be taken into account and the cross-section of the waveguide, relative to the optical wavelength, is not too big.
For multimode step index waveguides, in which the cross-section is significantly greater than the wavelength of the radiation used, efficient ray tracing based on geometrical optics is possible on the other hand.
In this case (in the simulation) a single ray of predetermined direction and polarization is coupled into the waveguide. This exits either directly at the end of the waveguide or is reflected on the wall of the optical channel, i.e. the wall surface of the index jump. This simulation is performed for a plurality of rays of different direction entering the waveguide. An example of this type is to be found in patent application DE 10051405 C2.
In this case the problem lies in determining the relevant reflection point. For simple geometrical forms, especially cuboids, this is easily possible, since the intersection point of a straight line with a plane is easy to calculate analytically. This is performed for the planes corresponding the side surfaces of the cuboid and that point which lies on the surface of the cuboid is used. This means however that only waveguides with a rectangular cross-section can be computed. This method is possible whenever there is a convex body with a surface which can be described analytically. However this is no longer the case for a curved waveguide.
To make curved waveguides with other cross sections accessible to computational determination, the waveguide is modeled as an extrusion of the cross-sectional surface, as for example in the Article by Th. Bierhoff and A. Himmler, “Modeling of board-integrated step index waveguides for advanced ray tracing analysis”, Proc. Optics in Computing Technologies, 33106 Paderborn 2001, P. 37-43. The cross-section is convex in this case; the extrusion along a curved trajectory however is no longer convex.
In practice even this method is not sufficient. The coupling of two waveguides by a molded material in particular produces a complex volume which can no longer be handled in the way mentioned. The same applies to the coupling-in points of waveguides. In this case concave areas with edges extending into them are produced in particular.